I think...
The situation is Windhelm is what happens everywhere when a lot of refugees of the same culture come to a new country, and this country isn't doing particularly well either, although way better than where the immigrants came from.
Considering the climate (and upbacked by in-game dialogue) it seems Windhelm would have some problems with poverty even without the Dunmer immigrants. Ant the civil war makes it worse. UIfric is annoyed with the Dunmer because he needs all his resources in the civil war, but it would be a smart move of him to win them over! The majority of the Dunmer don't like the Empire. OTOH Ambarys Rendar seems to be an important figure in the Gray Quarter, and his influence might be a problem here.
If the Imperials win the war, they might be interested in scoring political points by doing something for the Dunmer and trying to do something about the racial conflicts in the city. So if they're smart, yes, they will at least try to improve the situation.
If the Stowmcloaks win, Ulfric will probably put most energy into rebuilding the destroyed cities, so it could take a while before he does something about his own city.
*edited for typos